Properties

Degree $2$
Conductor $384$
Sign $0.577 + 0.816i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − i)3-s + (1.00 − 2.82i)9-s − 6i·11-s + 5.65i·17-s + 8.48·19-s − 5·25-s + (−1.41 − 5.00i)27-s + (−6 − 8.48i)33-s + 11.3i·41-s − 8.48·43-s + 7·49-s + (5.65 + 8.00i)51-s + (12 − 8.48i)57-s + 6i·59-s − 8.48·67-s + ⋯
L(s)  = 1  + (0.816 − 0.577i)3-s + (0.333 − 0.942i)9-s − 1.80i·11-s + 1.37i·17-s + 1.94·19-s − 25-s + (−0.272 − 0.962i)27-s + (−1.04 − 1.47i)33-s + 1.76i·41-s − 1.29·43-s + 49-s + (0.792 + 1.12i)51-s + (1.58 − 1.12i)57-s + 0.781i·59-s − 1.03·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.577 + 0.816i$
Motivic weight: \(1\)
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57036 - 0.812879i\)
\(L(\frac12)\) \(\approx\) \(1.57036 - 0.812879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.41 + i)T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 6iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 - 8.48T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 8.48T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 18iT - 83T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.36732314095345958866197170318, −10.18003742742986407974047232852, −9.203967438272745669856205301222, −8.310494733627895780293252179905, −7.71294832898232719335781053369, −6.43579796159215007122330409467, −5.59583465441700055415601694570, −3.79952816991674044677537430206, −2.97435160168075370410692631058, −1.27062072584849419492082993667, 2.02747850853675276182915285352, 3.29660434930338893539993961725, 4.53853832341200270929554884125, 5.35371331864920270843620974517, 7.22704519289053619322455979266, 7.54707509561433255840292960749, 8.953361458053080517317348205117, 9.703061591676662216923431894281, 10.17342159801229200023721588128, 11.55249471937628418532778875766

Graph of the $Z$-function along the critical line